Linear complementary dual (LCD) codes can be used to against side-channel attacks and fault noninvasive attacks. Let $d_{a}(n,6)$ and $d_{l}(n,6)$ be the minimum weights of all binary optimal linear codes and LCD codes with length $n$ and dimension 6, respectively.In this article, we aim to obtain the values of $d_{l}(n,6)$ for $n\geq 51$ by investigating the nonexistence and constructions of LCD codes with given parameters. Suppose that $s \ge 0$ and $0\leq t\leq 62$ are two integers and $n=63s+t$. Using the theories of defining vectors, generalized anti-codes, reduced codes and nested codes, we exactly determine $d_{l}(n,6)$ for $t \notin\{21,22,25,26,33,34,37,38,45,46\}$, while we show that $d_{l}(n,6)\in$$\{d_{a}(n,6)$ $-1,d_{a}(n,6)\}$ for $t\in\{21,22,26,34,37,38,46\}$ and $ d_{l}(n,6)\in$$ \{d_{a}(n,6)-2,$ $d_{a}(n,6)-1\}$ for$t\in{25,33,45\}$.

翻译：线性互补对偶（LCD）码可用于抵御侧信道攻击和故障非侵入攻击。令 $d_{a}(n,6)$ 和 $d_{l}(n,6)$ 分别表示长度为 $n$、维数为 6 的所有二进制最优线性码和 LCD 码的最小权重。本文旨在通过研究具有给定参数的 LCD 码的不存在性与构造，获得 $n\geq 51$ 时 $d_{l}(n,6)$ 的值。假设 $s \ge 0$ 和 $0\leq t\leq 62$ 是两个整数，且 $n=63s+t$。利用定义向量、广义反码、约化码和嵌套码的理论，我们精确确定了当 $t \notin\{21,22,25,26,33,34,37,38,45,46\}$ 时的 $d_{l}(n,6)$ 值；同时证明了当 $t\in\{21,22,26,34,37,38,46\}$ 时，$d_{l}(n,6)\in$$\{d_{a}(n,6)$ $-1,d_{a}(n,6)\}$；而当 $t\in\{25,33,45\}$ 时，$d_{l}(n,6)\in$$ \{d_{a}(n,6)-2,$ $d_{a}(n,6)-1\}$。